LEARN · EXPLAIN · REVISE

Prove n³−n+3tn is divisible by 3 for all n≥1.

Read the idea, work independently, then explain what changed.

高二選擇性必修第二册(A版).pdf · 4.4 · PDF 49 / printed page 44

TOPIC 01

Mathematical induction

Build proofs with a verified base, an explicit induction hypothesis and a valid step.

Build proofs with a verified base, an explicit induction hypothesis and a valid step.
Justify the method and check the conditions in a new situation.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Your turn

t=8

Induction proves a statement only for the specified integer domain starting at the base case.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Verify the base case, assume the claim at k, and prove it at k+1.

Hint 2

Use this intermediate relation.

f(k+1)−f(k)=3k(k+1)+3t

Worked solution

The requested relation or conclusion is shown below.

3 | (n^3−n+3tn)

Checks and common pitfalls: Both the base and the increment are multiples of three.

Reasoning checklist · self / teacher assessment

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

Build proofs with a verified base, an explicit induction hypothesis and a valid step.
Which condition is essential in mathematical induction?
Can a correct induction step rescue a false or missing base case?

Defining relation: Build proofs with a verified base, an explicit induction hypothesis and a valid step.
$P(n_0)\land[P(k)\Rightarrow P(k+1)]$
Conditions: Induction proves a statement only for the specified integer domain starting at the base case.